13.144. ROM144 - Reduced Order Electrostatic-Structural

Matrix or VectorShape Functions Integration Points
Stiffness MatrixNone (lumped)None
Damping MatrixNone (lumped)None
Mass MatrixNone (lumped)None
Load VectorNone (lumped)None

ROM144 represents a reduced order model of distributed electostatic-structural systems. The element is derived from a series of uncoupled static FEM analyses using electrostatic and structural elements (Reduced Order Modeling of Coupled Domains). The element fully couples the electrostatic-structural domains and is suitable for simulating the electromechanical response of micro-electromechanical systems (MEMS) such as clamped beams, micromirror actuators, and RF switches.

ROM144 is defined by either 20 (KEYOPT(1) = 0) or 30 nodes (KEYOPT(1) = 1). The first 10 nodes are associated with modal amplitudes, and represented by the EMF DOF labels. Nodes 11 to 20 have electric potential (VOLT) DOFs, of which only the first five are used. The last 10 optional nodes (21 to 30) have structural (UX) DOF to represent master node displacements in the operating direction of the device. For each master node, ROM144 internally uses additional structural DOFs (UY) to account for Lagrange multipliers used to represent internal nodal forces.

13.144.1. Element Matrices and Load Vectors

The FE equations of the 20-node option of ROM144 are derived from the system of governing equations of a coupled electrostatic-structural system in modal coordinates (Equation 14–149 and Equation 14–150)

(13–220)

where:

K = stiffness matrix
D = damping matrix
M = mass matrix
F = force
I = electric current

The system of Equation 13–220 is similar to that of the TRANS126 - Electromechanical Transducer element with the difference that the structural DOFs are generalized coordinates (modal amplitudes) and the electrical DOFs are the electrode voltages of the multiple conductors of the electromechanical device.

The contribution to the ROM144 FE matrices and load vectors from the electrostatic domain is calculated based on the electrostatic co-energy Wel (Reduced Order Modeling of Coupled Domains).

The electrostatic forces are the first derivative of the co-energy with respect to the modal coordinates:

(13–221)

where:

Fk = electrostatic force
Wel = co-energy
qk = modal coordinate
k = index of modal coordinate

Electrode charges are the first derivatives of the co-energy with respect to the conductor voltage:

(13–222)

where:

Qi = electrode charge
Vi = conductor voltage
i = index of conductor

The corresponding electrode current Ii is calculated as a time-derivative of the electrode charge Qi. Both, electrostatic forces and the electrode currents are stored in the Newton-Raphson restoring force vector.

The stiffness matrix terms for the electrostatic domain are computed as follows:

(13–223)

(13–224)

(13–225)

(13–226)

where:

l = index of modal coordinate
j = index of conductor

The damping matrix terms for the electrostatic domain are calculated as follows:

(13–227)

(13–228)

(13–229)

There is no contribution to the mass matrix from the electrostatic domain.

The contribution to the FE matrices and load vectors from the structural domain is calculated based on the strain energy WSENE (Reduced Order Modeling of Coupled Domains). The Newton-Raphson restoring force F, stiffness K, mass M, and damping matrix D are computed according to Equation 13–230 to Equation 13–233.

(13–230)

(13–231)

(13–232)

(13–233)

where:

i, j = indices of modal coordinates
ωi = angular frequency of ith eigenmode
ξi = modal damping factor (input as Damp on the RMMRANGE command

13.144.2. Combination of Modal Coordinates and Nodal Displacement at Master Nodes

For the 30-node option of ROM144, it is necessary to establish a self-consistent description of both modal coordinates and nodal displacements at master nodes (defined on the RMASTER command defining the generation pass) in order to connect ROM144 to other structural elements UX DOF or to apply nonzero structural displacement constraints or forces.

Modal coordinates qi describe the amplitude of a global deflection state that affects the entire structure. On the other hand, a nodal displacement ui is related to a special point of the structure and represents the true local deflection state.

Both modal and nodal descriptions can be transformed into each other. The relationship between modal coordinates qj and nodal displacements ui is given by:

(13–234)

where:

ϕij = jth eigenmode shape at node i
m = number of eigenmodes considered

Similarly, nodal forces Fi can be transformed into modal forces fj by:

(13–235)

where:

n = number of master nodes

Both the displacement boundary conditions at master nodes ui and attached elements create internal nodal forces Fi in the operating direction. The latter are additional unknowns in the total equation system, and can be viewed as Lagrange multipliers λi mapped to the UY DOF. Hence each master UX DOF requires two equations in the system FE equations in order to obtain a unique solution. This is illustrated on the example of a FE equation (stiffness matrix only) with 3 modal amplitude DOFs (q1, q2, q3), 2 conductors (V1, V2), and 2 master UX DOFs (u1, u2):

(13–236)

Rows 6 and 7 of Equation 13–236 correspond to the modal and nodal displacement relationship of Equation 13–234, while column 6 and 7 - to nodal and modal force relationship (Equation 13–235). Rows and columns (8) and (9) correspond to the force-displacement relationship for the UX DOF at master nodes:

(13–237)

(13–238)

where is set to zero by the ROM144 element. These matrix coefficients represent the stiffness caused by other elements attached to the master node UX DOF of ROM144.

13.144.3. Element Loads

In the generation pass of the ROM tool, the ith mode contribution factors for each element load case j (Reduced Order Modeling of Coupled Domains) are calculated and stored in the ROM database file. In the Use Pass, the element loads can be scaled and superimposed in order to define special load situations such as acting gravity, external acceleration or a pressure difference. The corresponding modal forces for the jth load case (Equation 14–149) is:

(13–239)

where:


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